Indices of Vector Fields on Real Analytic Varieties

AbstractLet V be an analytic variety in some open set in which contains the origin and which is purely k-dimensional. For a curve γ in , defined by a convergent Puiseux series and satisfying γ(0) = 0, and d ≥ 1, define Vt := t−d(V − (t)). Then the currents defined by Vt converge to a limit current Tγ,d[V] as t tends to zero. Tγ,d[V] is either zero or its support is an algebraic variety of pure dimension k in . Properties of such limit currents and examples are presented. These results will be applied in a forthcoming paper to derive necessary conditions for varieties satisfying the local Phragmén-Lindelöf condition that was used by Hörmander to characterize the constant coefficient partial differential operators which act surjectively on the space of all real analytic functions on .

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Normal forms for perturbations of systems possessing a Diophantine invariant torus

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.116 ◽

2017 ◽

Vol 39 (8) ◽

pp. 2176-2222 ◽

Cited By ~ 1

Author(s):

JESSICA ELISA MASSETTI

Keyword(s):

Vector Fields ◽

Normal Forms ◽

Invariant Tori ◽

Inverse Function ◽

Hamiltonian Mechanics ◽

Inverse Function Theorem ◽

Unified Framework ◽

Normal Form Theorem ◽

Real Analytic ◽

Twist Theorem

We give a new proof of Moser’s 1967 normal-form theorem for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. The proposed approach, based on an inverse function theorem in analytic class, is flexible and can be adapted to several contexts. This allows us to prove in a unified framework the persistence, up to finitely many parameters, of Diophantine quasi-periodic normally hyperbolic reducible invariant tori for vector fields originating from dissipative generalizations of Hamiltonian mechanics. As a byproduct, generalizations of Herman’s twist theorem and Rüssmann’s translated curve theorem are proved.

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MILNOR FIBRATIONS AND d-REGULARITY FOR REAL ANALYTIC SINGULARITIES

International Journal of Mathematics ◽

10.1142/s0129167x10006124 ◽

2010 ◽

Vol 21 (04) ◽

pp. 419-434 ◽

Cited By ~ 10

Author(s):

J. L. CISNEROS-MOLINA ◽

J. SEADE ◽

J. SNOUSSI

Keyword(s):

Fiber Bundle ◽

Critical Value ◽

Singular Set ◽

Small Sphere ◽

Milnor Fibration ◽

Analytic Varieties ◽

Real Analytic ◽

Analytic Singularities ◽

Analytic Maps ◽

Do So

We study Milnor fibrations of real analytic maps [Formula: see text], n ≥ p, with an isolated critical value. We do so by looking at a pencil associated canonically to every such map, with axis V = f-1(0). The elements of this pencil are all analytic varieties with singular set contained in V. We introduce the concept of d-regularity, which means that away from the axis each element of the pencil is transverse to all sufficiently small spheres. We show that if V has dimension 0, or if f has the Thom af-property, then f is d-regular if and only if it has a Milnor fibration on every sufficiently small sphere, with projection map f/‖f‖. Our results include the case when f has an isolated critical point. Furthermore, we show that if f is d-regular, then its Milnor fibration on the sphere is equivalent to its fibration on a Milnor tube. To prove these fibration theorems we introduce the spherefication map, which is rather useful to study Milnor fibrations. It is defined away from V; one of its main properties is that it is a submersion if and only if f is d-regular. Here restricted to each sphere in ℝn the spherefication gives a fiber bundle equivalent to the Milnor fibration.

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Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

Journal of Dynamical and Control Systems ◽

10.1007/s10883-016-9314-y ◽

2016 ◽

Vol 23 (1) ◽

pp. 89-109 ◽

Cited By ~ 3

Author(s):

Jessica Angélica Jaurez-Rosas

Keyword(s):

Vector Fields ◽

Real Plane ◽

Analytic Vector ◽

The Real ◽

Real Analytic

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